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Welded Beam Design Optimization

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Welded Beam Design Optimization


The diagram illustrates a rigid member welded onto a beam. A load is applied to the end of the member.



The beam is to be optimized for minimum cost by varying the weld and member dimensions x1, x2, x3 and x4. The constraints include limits on the shear stress, bending stress, buckling load and end deflection. The variables x1 and x2 are usually integer multiples of 0.0625 inch, but for this application are assumed continuous.


Reference: K. Ragsdell and D. Phillips. Optimal Design of a Class of Welded Structures using Geometric Programming. J. Eng. Ind., 98(3):1021–1025, 1976.



Young's modulus (psi)

E := 30*10^6:

Shearing modulus for the beam material (psi)

G := 12*10^6:

Overhang length of the member (inch)

L := 14:

Design stress of the weld (psi)

`τ__max` := 13600:

Design normal stress for the beam material (psi)

`σ__max` := 30000:

Maximum deflection (inch)

`δ__max` := .25:

Load (lb)

P := 6000:

Cost per unit volume of the weld material ($ inch)

C__1 := .10471:

Cost per unit volume of the bar ($ inch)

C__2 := 0.4811e-1:

Labor cost per unit weld volume ($ inch)

C__3 := 1:

Cost Function

Volume of weld material (inch^3)

V__weld := x__1^2*x__2:

Volume of bar (inch^3)

V__bar := x__3*x__4*(L+x__2):

Total material cost to be minimized.

f := proc (x__1, x__2, x__3, x__4) options operator, arrow; (C__1+C__3)*V__weld+C__2*V__bar end proc:


The shear stress at the beam support location cannot exceed the maximum allowable for the material

con1 := `τ__max`-tau(x__1, x__2, x__3, x__4) >= 0:

The normal bending stress at the beam support location cannot exceed the maximum yield strength for the material

con2 := `σ__max`-sigma(x__1, x__2, x__3, x__4) >= 0:

The member thickness is greater than the weld thickness

con3 := x__4-x__1 >= 0:

con4 := C__1*x__1^2+C__2*x__3*x__4*(L+x__2)-5 <= 0:

The weld thickness must be larger than a defined minimum

con5 := x__1-.125 >= 0:

The deflection cannot exceed the maximum deflection

con6 := `&delta;__max`-delta(x__1, x__2, x__3, x__4) >= 0:

The buckling load is greater than the applied load

con7 := Pc(x__1, x__2, x__3, x__4)-P >= 0:

Size constraints

con8 := x__1 >= .1, x__4 <= 2.0, x__2 >= .1, x__3 <= 10:

Collect all the constraints

cons := {con1, con2, con3, con4, con5, con6, con7, con8}:

Engineering Relationships

Weld stress

tau := proc (x__1, x__2, x__3, x__4) options operator, arrow; sqrt(`&tau;__d`^2+`&tau;__d`*`&tau;__dd`*x__2/R+`&tau;__dd`^2) end proc:

Primary stress acting over the weld throat

`&tau;__d` := P/(sqrt(2)*x__1*x__2):

Secondary torsional stress.

`&tau;__dd` := M*R/J:

Moment of P about center of gravity of weld setup

M := P*(L+(1/2)*x__2):

R := sqrt((1/4)*x__2^2+((x__1+x__3)*(1/2))^2):

Polar moment of inertia of weld

J := 2*x__1*x__2*sqrt(2)*((1/12)*x__2^2+((x__1+x__3)*(1/2))^2):

Bar bending stress

sigma := proc (x__1, x__2, x__3, x__4) options operator, arrow; 6*P*L/(x__4*x__3^2) end proc:

Bar Deflection. To calculate the deflection, assume the bar to be a cantilever of length L

delta := proc (x__1, x__2, x__3, x__4) options operator, arrow; 4*P*L^3/(E*x__4*x__3^2) end proc:

For narrow rectangular bars, the bar buckling load is approximated by (Timoshenko, S., and J. Gere,Theory of Elastic Stability,McGraw-Hill, New York, 1961, p. 257)

Pc := proc (x__1, x__2, x__3, x__4) options operator, arrow; 4.013*E*sqrt((1/36)*x__3^2*x__4^6)*(1-(1/2)*x__3*sqrt((1/4)*E/G)/L)/L^2 end proc:


bounds := x__1 = 0 .. 10, x__2 = 0 .. 10, x__3 = 0 .. 10, x__4 = 0 .. 10:

Hence the minimum cost and optimized dimensions are

sol := Optimization:-Minimize(f(x__1, x__2, x__3, x__4), cons, bounds)

[1.72485230854216631, [x__1 = HFloat(0.20572963977072647), x__2 = HFloat(3.4704886658286433), x__3 = HFloat(9.036623910694827), x__4 = HFloat(0.20572963977072622)]]